Projectile Motion¶
In this part, we study the general motion in two dimensions near to the surface of earth. This motion is commonly seen in our lives, for example can occur in a baseball or soccer game, when we played basketball and shoot the ball. These examples are projectile motion, now we focus on the process through the object is projected, only consider the motion after projected and before to touch the surface (1). Note: for now, we deprecated the air resistance.
Galileo was the first to describe the motion of projectiles in the right way. He showed that this motion can be understood through rectangular components of velocity $\vec{v}$, this mean that, if separate the velocity vector in a horizontal $\vec{v}_{x}$ and vertical $\vec{v}_{y}$ components we can analyze motion by separated.
The figure shows a projectile motion, projected horizontally, by convenience it assumes that the motion starts at the origin of $xy$ plane at $t=0$ , therefore the rectangular components are $x_{0}=y_{0}=0$.
Taking advantage of vector analysis, we can see the evolution of projectile motion. If an object, it is projected upwards with an angle, the analysis is similar with an exception to velocity, in this case the exists a vertical component or rather a $v_{y0}$.
The equations in terms of the horizontal and vertical components of the projectile with constant acceleration in two dimensions are :
\begin{align} v_x &= v_{x0} + a_{x}t & v_{y}&= v_{y0} + a_{y}t \\ x &= x_{0}+v_{x0}+\dfrac{1}{2}a_{x}t^{2} & y&= y_{0}+v_{y0}+\dfrac{1}{2}a_{y}t^{2}\\ v_{x}^{2} &= v^{2}_{x0} + 2a_{x}(x-x_{0}) & v_{y}^{2} &= v^{2}_{y0} + 2a_{y}(y-y_{0}) \end{align}Also, this is possible to simplification this equations if consider that $a_{x}=0$ for projectile motion, and also we suppose that $a_{y} = -g = -9.80\mathrm{m/s^{2}}$, finally if we take the angle in relation with $+x$ axis, we have :
\begin{equation} \begin{split} v_{x0} = v_{0} \cos \theta\\ v_{y0} = v_{0} \sin \theta \end{split} \end{equation}Work out problems about of projectile motion involves that in this motion should be considered the time, the object travel into air only influenced by the gravity. For this case, is necessary only consider the gravity effects in the motion object, so, it can establish $\vec{a}=\vec{g}$. The equations of projectile motion then
\begin{align} v_x &= v_{x0} & v_{y}&= v_{y0} -gt \\ x &= x_{0} + v_{x0}t & y&= y_{0}+v_{y0}t-\dfrac{1}{2}gt^{2}\\ & & v_{y}^{2} &= v^{2}_{y0} + 2g(y-y_{0}) \end{align}